[Paper Review] Dynamical charged black hole spontaneous scalarization in Anti-de Sitter spacetimes

Cheng-Yong Zhang, Peng Liu+3 more|arXiv (Cornell University)|Mar 25, 2021

Black Holes and Theoretical Physics118 references32 citations

TL;DR

This paper investigates the fully nonlinear dynamical evolution of charged black holes in Anti-de Sitter (AdS) spacetimes within Einstein-Maxwell-scalar theory with a coupling function f(φ) = e⁻ᵇφ². It demonstrates that sufficiently large black hole charge Q and coupling constant −b, combined with a small cosmological constant |Λ|, trigger spontaneous scalarization, leading to the formation of hairy black holes with exponentially growing irreducible mass that eventually saturates. The key contribution is the confirmation of phase structure consistency between nonlinear dynamics and eigenvalue analysis.

ABSTRACT

We study the fully nonlinear dynamics of black hole spontaneous scalarizations in Einstein-Maxwell scalar theory with coupling function $f(\phi)=e^{-b\phi^{2}}$, which can transform usual Reissner-Nordstr\"om Anti-de Sitter (RN-AdS) black holes into hairy black holes. Fixing the Arnowitt-Deser-Misner mass of the system, the initial scalar perturbation will destroy the original RN-AdS black hole and turn it into a hairy black hole provided that the constant $-b$ in the coupling function and the charge of the original black hole are sufficiently large, while the cosmological constant is small enough. In the scalarization process, we observe that the black hole irreducible mass initially increases exponentially, then it approaches to and finally saturates at a finite value. Choosing stronger coupling and larger black hole charge, we find that the black hole mass exponentially grows earlier and it takes a longer time for a hairy black hole to be developed and stabilized. We further examine phase structure properties in the scalarization process and confirm the observations in the non-linear dynamical study.

Motivation & Objective

• To study the fully nonlinear dynamical evolution of spontaneous scalarization in Einstein-Maxwell-scalar theory in asymptotically AdS spacetimes.
• To determine the conditions under which Reissner-Nordström-AdS (RN-AdS) black holes evolve into hairy black holes via scalar field instability.
• To examine the role of coupling strength (−b), black hole charge (Q), and cosmological constant (Λ) in triggering and shaping the scalarization process.
• To confirm the consistency between nonlinear dynamical results and phase structure analysis based on eigenvalue problems.

Proposed method

• Numerical solution of the coupled Einstein-Maxwell-scalar equations with f(φ) = e⁻ᵇφ² in spherical symmetry.
• Implementation of boundary conditions at spatial infinity and the black hole horizon for scalar and gauge fields.
• Use of a pseudospectral method to evolve the system in time, tracking the scalar field amplitude and black hole mass.
• Analysis of the scalar field's dominant damping mode to assess stability and timescale of hair formation.
• Phase structure analysis via eigenvalue problem to identify critical lines separating RN-AdS and hairy black hole phases.
• Comparison of dynamical evolution with phase structure predictions to validate findings.

Experimental results

Research questions

• RQ1 Under what conditions does a charged RN-AdS black hole undergo spontaneous scalarization in the presence of a non-minimal coupling f(φ) = e⁻ᵇφ²?
• RQ2 How does the growth of the scalar hair and the increase in black hole irreducible mass evolve over time in the nonlinear regime?
• RQ3 What is the dependence of the scalarization onset time and saturation timescale on the coupling constant −b and black hole charge Q?
• RQ4 How do the cosmological constant Λ and the coupling parameter b influence the critical threshold for scalarization?
• RQ5 To what extent do the phase structure predictions based on eigenvalue analysis match the outcomes of nonlinear dynamical simulations?

Key findings

• For fixed ADM mass M = 1, sufficiently large Q and −b, combined with small |Λ|, trigger spontaneous scalarization, transforming RN-AdS black holes into hairy black holes.
• The black hole irreducible mass increases exponentially at early times and then saturates at a finite final value, with the growth rate γi decreasing and saturation rate γf increasing as |Λ| increases.
• Larger Q and stronger coupling (larger −b) lead to earlier onset of scalar hair growth and longer times to stabilization.
• The final scalar field amplitude φf changes smoothly near critical Λ∗, but unsmoothly near critical b∗ and Q∗, indicating a phase transition.
• The phase structure analysis confirms that scalarization is more easily triggered in asymptotically flat spacetimes than in AdS spacetimes, contrary to linear perturbation expectations.
• The dominant damping mode's imaginary part ωI tends to zero for larger Q and stronger coupling, indicating longer stabilization times.